Problem 121 - Elementary and Intermediate Algebra

Question

If \(\$ P\) is invested at the end of each year in an annuity earning annual interest at a rate \(r,\) the amount in the account will be \(\$ A\) after \(n\) years, where $$n=\frac{\log \left(\frac{A r}{P}+1\right)}{\log (1+r)}$$ If \(\$ 1,000\) is invested each year in an annuity earning \(12 \%\) annual interest, how long will it take for the account to be worth \(\$ 20,000 ?\) Round to the nearest tenth of a year.

Step-by-Step Solution

Answer

It will take approximately 10.8 years.

Step 1: Understand the Given Equation

Given the formula \( n = \frac{\log \left(\frac{A r}{P}+1\right)}{\log (1+r)} \), we need to identify the components: \( P = 1000 \), the annual investment; \( r = 0.12 \), the interest rate; and \( A = 20000 \), the account value after \( n \) years.

Step 2: Substitute the Known Values

Substitute the known values into the formula: \( n = \frac{\log \left(\frac{20000 \times 0.12}{1000}+1\right)}{\log (1+0.12)} \).

Step 3: Simplify the Numerator

Calculate \( \frac{20000 \times 0.12}{1000} + 1 = 2.4 + 1 = 3.4 \). So, the numerator of the fraction is \( \log(3.4) \).

Step 4: Simplify the Denominator

Calculate \( 1+0.12 = 1.12 \). Therefore, the denominator of the fraction is \( \log(1.12) \).

Step 5: Compute the Logs

Find \( \log(3.4) \approx 0.5315 \) and \( \log(1.12) \approx 0.0492 \).

Step 6: Calculate the Number of Years

Divide the logarithm from the numerator by the logarithm from the denominator: \( n \approx \frac{0.5315}{0.0492} \approx 10.8 \).

Step 7: Round to the Nearest Tenth

Since the result is \( 10.8 \), it remains \( 10.8 \) when rounded to the nearest tenth.

Key Concepts

Logarithms in Finance

Logarithms are mathematical tools that can simplify complex calculations, especially when dealing with exponential growth scenarios, such as financial growth over time.
In finance, logarithms are often used to solve for unknown variables in formulas involving compound interest and annuities.
They help to find the length of time it takes for an investment to reach a certain value.
In the given exercise, the formula \(n = \frac{\log \left(\frac{Ar}{P}+1\right)}{\log (1+r)}\) is used to determine how long it will take for an annuity to reach a target amount.
The logarithm in the numerator helps us find out how much the investment grows in relation to the initial amount and the interest rate over time.

• Numerator: \(\log \left(\frac{Ar}{P}+1\right)\)
• Denominator: \(\log(1+r)\)

This means the numerator calculates the total growth required and the denominator gives us the effect of one year's growth. Logarithms thus transform the exponential growth situation into a simpler linear form, allowing for easy computation using basic arithmetic.

Annual Interest Rate

The annual interest rate plays a crucial role in determining how quickly an investment will grow over time.
In our context, the interest rate, denoted as \( r \), is given as 12% or 0.12 when expressed as a decimal.
The annual interest rate influences both the amount of growth within the annuity and the time taken to reach the target amount.To break it down:

• It represents the percentage of the principal amount that will be earned or paid in interest over one year.
• Expressed in the formula as \( 1 + r \), it helps calculate the growth multiplier.
• High annual interest rates lead to a faster accumulation of money in an annuity.

Considering an example: investing \(\$1000\) annually with an interest rate of 12% means each year's investment grows by that percentage during the year.
Therefore, understanding how \( r \) impacts your investments can help plan savings and investments effectively.

Investment Growth

Investment growth in annuities follows an interesting pattern due to the power of compound interest.
It refers to how your money expands over time in an account, affected by the amount invested, the rate of interest, and the duration for which the money is invested.Key factors of investment growth include:

• The principal amount, \( P \), is what you invest at regular intervals.
• The time, \( n \), determines how long you're investing, indicating how many periods your money has to grow.
• The formula as used in the exercise looks at accumulated interest over multiple cycles, compounding your investment.

For example, with \( \\(1000 \) invested at the end of each year at 12% interest, it is not just the \( \\)1000 \) that earns interest.
The interest added in each previous year also earns interest, leading to a growth pattern that is exponential rather than linear.
Thus, the longer the investment period, the larger the impact of compounding, and hence more substantial the growth.